Differential Equations

JEE Concepts

Order and Degree of a Differential Equation:

• The order of a differential equation is the highest order of the derivative that appears in the equation.
• The degree of a differential equation is the highest power to which the highest-order derivative is raised.

Linear and Non-linear Differential Equations:

• A differential equation is linear if it can be written in the form: $$a_ny^{(n)}+a_{n-1}y^{(n-1)}+…+a_1y’+a_0y=b$$ where (a_n\ne 0) and (b) are constants.
• A differential equation is non-linear if it cannot be written in the form of a linear differential equation.

Homogeneous and Non-homogeneous Differential Equations:

• A differential equation is homogeneous if it can be written in the form: $$a_ny^{(n)}+a_{n-1}y^{(n-1)}+…+a_1y’+a_0y=0$$ where (a_n, a_{n-1}, …, a_1, a_0) are constants.
• A differential equation is non-homogeneous if it cannot be written in the form of a homogeneous differential equation.

Exact Differential Equations and Integrating Factors:

• An exact differential equation is a differential equation that can be written in the form: $$M(x,y)dx+N(x,y)dy=0$$ where (M(x,y)) and (N(x,y)) have continuous first-order partial derivatives in some region.
• An integrating factor for an exact differential equation is a function (I(x,y)) such that the product (I(x,y)M(x,y)dx + I(x,y)N(x,y)dy) is an exact differential.

Separation of Variables:

• The method of separation of variables can be used to solve some first-order differential equations.
• To solve a first-order differential equation using separation of variables, we write the equation in the form: $$\frac{dy}{dx}=f(x)g(y)$$ and then integrate both sides of the equation.

Linear Differential Equations of First Order and First Degree:

• A linear differential equation of first order and first degree is an equation that can be written in the form: $$\frac{dy}{dx}+P(x)y=Q(x)$$ where (P(x)) and (Q(x)) are continuous functions in some interval.
• This type of differential equation can be solved using an integrating factor.

Bernoulli’s Differential Equation:

• A Bernoulli differential equation is a differential equation that can be written in the form: $$\frac{dy}{dx}+P(x)y=Q(x)y^n$$ where (n) is a constant.
• This type of differential equation can be transformed into a linear differential equation by making the substitution (v=y^{1-n}).

Riccati’s Differential Equation:

• A Riccati differential equation is a differential equation that can be written in the form: $$\frac{dy}{dx}=P(x)y^2+Q(x)y+R(x)$$ where (P(x), Q(x)), and (R(x)) are continuous functions in some interval.
• This type of differential equation is difficult to solve in general, but there are some special cases that can be solved using a variety of techniques.

Exact Differential Equations

• An exact differential equation is one for which the integrating factor is (1).
• To check if a differential equation is exact, we first compute its total differential. If it is equal to zero, then the equation is exact.

Linear Differential Equations of Higher Order with Constant Coefficients:

• A linear differential equation of higher order with constant coefficients is an equation that can be written in the form: $$a_ny^{(n)}+a_{n-1}y^{(n-1)}+…+a_1y’+a_0y=0$$ where (a_n, a_{n-1}, …, a_1, a_0) are constants.
• This type of differential equation can be solved using the method of undetermined coefficients or the method of variation of parameters.

Variation of Parameters:

• Variation of parameters is a method for solving non-homogeneous linear differential equations of higher order with constant coefficients.
• The method involves replacing the constants in the homogeneous solution with functions of (x) and then solving the resulting system of differential equations.

Cauchy-Euler Equation:

• A Cauchy-Euler equation is a linear differential equation of second order with variable coefficients that can be written in the form: $$x^2y’’+ax y’ +by=0$$ where (a) and (b) are constants.
• This type of differential equation can be solved by making the substitution (x=e^t).

Legendre’s Equation:

• Legendre’s equation is a linear differential equation of second order with variable coefficients that can be written in the form: $$(1-x^2)y’’ - 2xy’+\nu (\nu + 1)y=0$$ where (\nu) is a constant.
• This type of differential equation is important in the study of spherical harmonics.

Bessel’s Equation:

• Bessel’s equation is a linear differential equation of second order with variable coefficients that can be written in the form: $$x^2y’’+xy’+(x^2-\nu^2)y=0$$ where (\nu) is a constant.
• This type of differential equation is important in the study of Bessel functions.

Frobenius Method:

• The Frobenius method is a method for solving linear differential equations with regular singular points.
• The method involves making a series expansion of the solution around the singular point and then determining the coefficients of the series.

Power Series Solutions:

• Power series solutions are solutions to differential equations that can be expressed as a power series.

• Power series solutions are often used to solve differential equations that have regular singular points.

                        **CBSE Concepts**
  

Order and Degree of a Differential Equation:

• The order of a differential equation is the highest order of the derivative that appears in the equation.
• The degree of a differential equation is the highest power to which the highest-order derivative is raised.

Linear and Non-linear Differential Equations:

• A differential equation is linear if it can be written in the form: $$a_ny^{(n)}+a_{n-1}y^{(n-1)}+…+a_1y’+a_0y=b$$ where (a_n\ne 0) and (b) are constants.
• A differential equation is non-linear if it cannot be written in the form of a linear differential equation.

Homogeneous and Non-homogeneous Differential Equations:

• A differential equation is homogeneous if it can be written in the form: $$a_ny^{(n)}+a_{n-1}y^{(n-1)}+…+a_1y’+a_0y=0$$ where (a_n, a_{n-1}, …, a_1, a_0) are constants.
• A differential equation is non-homogeneous if it cannot be written in the form of a homogeneous differential equation.

Exact Differential Equations and Integrating Factors:

• An exact differential equation is a differential equation that can be written in the form: $$M(x,y)dx+N(x,y)dy=0$$ where (M(x,y)) and (N(x,y)) have continuous first-order partial derivatives in some region.
• An integrating factor for an exact differential equation is a function (I(x,y)) such that the product (I(x,y)M(x,y)dx + I(x,y)N(x,y)dy) is an exact differential.

Separation of Variables:

• The method of separation of variables can be used to solve some first-order differential equations.
• To solve a first-order differential equation using separation of variables, we write the equation in the form: $$\frac{dy}{dx}=f(x)g(y)$$ and then integrate both sides of the equation.

Linear Differential Equations of First Order and First Degree:

• A linear differential equation of first order and first degree is an equation that can be written in the form: $$\frac{dy}{dx}+P(x)y=Q(x)$$ where (P(x)) and (Q(x)) are continuous functions in some interval.
• This type of differential equation can be solved using an integrating factor.

Bernoulli’s Differential Equation:

• A Bernoulli differential equation is a differential equation that can be written in the form: $$\frac{dy}{dx}+P(x)y=Q(x)y^n$$ where (n) is a constant.
• This type of differential equation can be transformed into a linear differential equation by making the substitution (v=y^{1-n}).

Riccati’s Differential Equation:

• A Riccati differential equation is a differential equation that can be written in the form: $$\frac{dy}{dx}=P(x)y^2